Some Application Of DarcyWeisbach Formula In Steady Pipe FlowsII
M.S. Ghidaoui (spring 2002)
In this set of problems, both frictional losses and local (minor losses) are considered. In
addition, both simple and branched pipe systems are analyzed in this chapter. In all the
problems below, unless stated otherwise, assume water temperature is 15
°
C
(
v
= 1.13 x
10
6
m
2
/S).
Problem 1:
A pipeline 20 km long delivers water from an impounding reservoir to a
service reservoir the minimum difference in level between which is 100 m.
The pipe of
uncoated cast iron (
ε
= 0.3 mm) is 400 mm in diameter.
Local losses, including entry
loss and velocity head amount to 10 V
2
/2g.
1.
Calculate the minimum uncontrolled discharge to the service reservoir.
2.
What additional head loss would need to be created by a valve to regulate the
discharge to 160 l/s?
Solution:
H
1
2
1): The energy equation is:
g
V
g
V
D
fL
g
V
Z
P
g
V
Z
P
2
2
10
2
2
2
2
2
2
2
2
1
2
1
1
+
+
+
+
=
+
+
γ
γ
Where,
f (L/D)( v
2
/2g )
~ Friction Losses
10 v
2
/2g
~ Minor (local) Losses
P
1
= P
2
= 0; v
1
≈
v
2
≈
0
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g
V
g
V
D
fL
H
Z
Z
2
2
10
2
2
2
1
+
=
=

Or
ε
/D= 0.3/400 =
0.00075;
Assume
V= 2m/s
⇒
Re= 7.27*10
5
⇒
From Moody chart:
f=0.019
Using this f check whether or not the energy equation is satisfied:
V
min
= (2g*100)/
(10+0.019*20,000/0.4) = 1.43 m/s
The value assumed is 2 m/s but the value needed to satisfy the energy equation is 1.43
m/s. Not acceptable and we need to try again.
Assume V=1.43 m/s
⇒
New Re = 5.2*10
9
⇒
From Moody chart: f = 0.0192
Using f=0.0192 check whether or not the energy equation is satisfied:
→
We
obtain :
V
min
=1.422 m/s
→
The value assumed is 1.43 m/s and the value needed to satisfy the
energy equation is 1.422 m/s. Close enough, we accept 1.422 m/s as our
solution for
velocity. The flowrate associated with this velocity is:
Q
min
≈
179 l/s
2):
Q = 160 l/s
→
V = 1.273m/s
→
Re = 4.5*10
9
→
f = 0.0193
(
29
m
h
v
g
h
h
g
V
g
V
D
fL
v
v
4
.
19
273
.
1
2
1
10
4
.
0
000
,
20
0193
.
0
100
2
10
2
100
2
2
2
=
+
×

=
+
+
=
where h
v
is additional valve losses
Problem 2:
A long, straight horizontal pipeline of diameter 350 mm and effective
roughness size 0.03 mm is to be constructed to convey crude oil of density 860 kg/m3
and absolute viscosity 0.0064 Ns/m2 from the oilfield to a port at a steady rate of 7000
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 Spring '10
 KK
 Fluid Dynamics, energy equation, Service reservoir, effective roughness size

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